| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.CycleCalculus |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Filter Set |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| local notation "BL" => boundedLipSubmodule (X := X) |
| noncomputable def mixedTuple {ι : Type*} [LinearOrder ι] {E : ι → Type*} |
| (a b : ∀ i, E i) (i : ι) : ∀ j, E j := fun j => if j < i then a j else b j |
|
|
| lemma MultilinearMap.sub_eq_sum_update {ι : Type*} [Fintype ι] [LinearOrder ι] |
| {E : ι → Type*} [∀ i, AddCommGroup (E i)] [∀ i, Module ℝ (E i)] |
| (M : MultilinearMap ℝ E ℝ) (a b : ∀ i, E i) : |
| M a - M b = ∑ i, M (Function.update (mixedTuple a b i) i (a i-b i)) := by |
| have h := M.map_sub_map_piecewise a b Finset.univ |
| simp only [Finset.piecewise_univ,Finset.mem_univ,true_implies] at h |
| rw [h] |
| apply Finset.sum_congr rfl |
| intro i _ |
| congr 1 |
| funext j |
| by_cases hji : j=i |
| · subst j |
| simp |
| · by_cases hj : j < i <;> simp [mixedTuple,hj,hji,Ne.symm hji] |
|
|
| lemma IsMetricCurrent.cycle_difference {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (a b : Fin (k+2) → BL) : |
| hcur.fullMultilinear a - hcur.fullMultilinear b = |
| ∑ i, (-1:ℝ)^i.val * T (a i-b i) |
| (fun j => ((i.removeNth (mixedTuple a b i)) j : X → ℝ)) := by |
| rw [MultilinearMap.sub_eq_sum_update] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact hcur.fullMultilinear_cycle_update hT hz _ _ _ |
|
|
| lemma IsMetricCurrent.cycle_difference_bound {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (a b : Fin (k+2) → BL) (K : ℝ≥0) |
| (ha : ∀ i, LipschitzWith K (a i : X → ℝ)) |
| (hb : ∀ i, LipschitzWith K (b i : X → ℝ)) |
| (D : ℝ) (hd : ∀ i x, |(a i : X → ℝ) x - (b i : X → ℝ) x| ≤ D) : |
| |hcur.fullMultilinear a - hcur.fullMultilinear b| ≤ |
| ((k+2:ℕ):ℝ) * (K:ℝ)^(k+1) * mass T * D := by |
| rw [hcur.cycle_difference hT hz] |
| calc |
| |∑ i, (-1:ℝ)^i.val * T (a i-b i) |
| (fun j => ((i.removeNth (mixedTuple a b i)) j : X → ℝ))| ≤ |
| ∑ i : Fin (k+2), (K:ℝ)^(k+1) * mass T * D := by |
| refine (Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum fun i _ => ?_) |
| simp only [abs_mul,abs_pow,abs_neg,abs_one,one_pow,one_mul] |
| have H := hcur.mass_bound_uniform (a i-b i).property (fun _ => K) |
| (fun j => show LipschitzWith K |
| ((i.removeNth (mixedTuple a b i)) j : X → ℝ) from by |
| dsimp [Fin.removeNth,mixedTuple] |
| split_ifs <;> first | exact ha _ | exact hb _) |
| (hd i) |
| simpa only [Finset.prod_const,Finset.card_univ,Fintype.card_fin,mul_assoc, |
| Submodule.coe_sub,Pi.sub_apply,mul_comm D] using H |
| _ = _ := by simp [mul_assoc] |
|
|
| end CAT0Fillings |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| local notation "BL" => boundedLipSubmodule (X := X) |
|
|
| lemma IsMetricCurrent.cycle_time_lipschitz {k : ℕ} {T : Functional X (k+1)} |
| (hcur : IsMetricCurrent T) (hT : IntegerRectifiable T) (hz : IsCycle T) |
| (V : ℝ → Fin (k+2) → BL) (K : ℝ≥0) |
| (hx : ∀ t i, LipschitzWith K (V t i : X → ℝ)) |
| (ht : ∀ i x, LipschitzWith K (fun t => (V t i : X → ℝ) x)) : |
| LipschitzWith (⟨((k+2:ℕ):ℝ) * (K:ℝ)^(k+1) * mass T * (K:ℝ),by |
| exact mul_nonneg (mul_nonneg (mul_nonneg (by positivity) (by positivity)) (mass_nonneg T)) K.coe_nonneg⟩ : ℝ≥0) (fun t => hcur.fullMultilinear (V t)) := by |
| apply lipschitzWith_iff_dist_le_mul.mpr |
| intro t s |
| have H := hcur.cycle_difference_bound hT hz (V t) (V s) K (hx t) (hx s) |
| ((K:ℝ)*dist t s) (fun i x => by simpa only [Real.dist_eq] using (ht i x).dist_le_mul t s) |
| change |hcur.fullMultilinear (V t) - hcur.fullMultilinear (V s)| ≤ |
| ((((k+2:ℕ):ℝ) * (K:ℝ)^(k+1) * mass T) * (K:ℝ)) * dist t s |
| simpa only [mul_assoc] using H |
|
|
| end CAT0Fillings |
| end |
| end |
|
|
| end OAI |
|
|
